\(\int \frac {1}{(1-i \sinh (c+d x))^4} \, dx\) [63]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [B] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 14, antiderivative size = 117 \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=-\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))} \]

[Out]

-1/7*I*cosh(d*x+c)/d/(1-I*sinh(d*x+c))^4-3/35*I*cosh(d*x+c)/d/(1-I*sinh(d*x+c))^3-2/35*I*cosh(d*x+c)/d/(1-I*si
nh(d*x+c))^2-2/35*I*cosh(d*x+c)/d/(1-I*sinh(d*x+c))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2729, 2727} \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4} \]

[In]

Int[(1 - I*Sinh[c + d*x])^(-4),x]

[Out]

((-1/7*I)*Cosh[c + d*x])/(d*(1 - I*Sinh[c + d*x])^4) - (((3*I)/35)*Cosh[c + d*x])/(d*(1 - I*Sinh[c + d*x])^3)
- (((2*I)/35)*Cosh[c + d*x])/(d*(1 - I*Sinh[c + d*x])^2) - (((2*I)/35)*Cosh[c + d*x])/(d*(1 - I*Sinh[c + d*x])
)

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
 x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}+\frac {3}{7} \int \frac {1}{(1-i \sinh (c+d x))^3} \, dx \\ & = -\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}+\frac {6}{35} \int \frac {1}{(1-i \sinh (c+d x))^2} \, dx \\ & = -\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}+\frac {2}{35} \int \frac {1}{1-i \sinh (c+d x)} \, dx \\ & = -\frac {i \cosh (c+d x)}{7 d (1-i \sinh (c+d x))^4}-\frac {3 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^3}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))^2}-\frac {2 i \cosh (c+d x)}{35 d (1-i \sinh (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=\frac {-21 i \cosh \left (\frac {3}{2} (c+d x)\right )+i \cosh \left (\frac {7}{2} (c+d x)\right )+35 \sinh \left (\frac {1}{2} (c+d x)\right )-7 \sinh \left (\frac {5}{2} (c+d x)\right )}{70 d \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^7} \]

[In]

Integrate[(1 - I*Sinh[c + d*x])^(-4),x]

[Out]

((-21*I)*Cosh[(3*(c + d*x))/2] + I*Cosh[(7*(c + d*x))/2] + 35*Sinh[(c + d*x)/2] - 7*Sinh[(5*(c + d*x))/2])/(70
*d*(Cosh[(c + d*x)/2] - I*Sinh[(c + d*x)/2])^7)

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.44

method result size
risch \(-\frac {4 \left (-7 \,{\mathrm e}^{d x +c}+21 i {\mathrm e}^{2 d x +2 c}+35 \,{\mathrm e}^{3 d x +3 c}-i\right )}{35 d \left ({\mathrm e}^{d x +c}+i\right )^{7}}\) \(51\)
derivativedivides \(\frac {\frac {16 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {72}{5 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {12}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {16}{7 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {8 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {6 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) \(121\)
default \(\frac {\frac {16 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {72}{5 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {12}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {16}{7 \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {8 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {6 i}{\left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{d}\) \(121\)
parallelrisch \(\frac {-\frac {12}{35}+\frac {12 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35}-\frac {6 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5}+\frac {2 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}+\frac {6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5}}{d \left (-21 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+7 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+35 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-35 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-7 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+21 i \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-i\right )}\) \(169\)

[In]

int(1/(1-I*sinh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

-4/35*(-7*exp(d*x+c)+21*I*exp(2*d*x+2*c)+35*exp(3*d*x+3*c)-I)/d/(exp(d*x+c)+I)^7

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=-\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} + 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} - i\right )}}{35 \, {\left (d e^{\left (7 \, d x + 7 \, c\right )} + 7 i \, d e^{\left (6 \, d x + 6 \, c\right )} - 21 \, d e^{\left (5 \, d x + 5 \, c\right )} - 35 i \, d e^{\left (4 \, d x + 4 \, c\right )} + 35 \, d e^{\left (3 \, d x + 3 \, c\right )} + 21 i \, d e^{\left (2 \, d x + 2 \, c\right )} - 7 \, d e^{\left (d x + c\right )} - i \, d\right )}} \]

[In]

integrate(1/(1-I*sinh(d*x+c))^4,x, algorithm="fricas")

[Out]

-4/35*(35*e^(3*d*x + 3*c) + 21*I*e^(2*d*x + 2*c) - 7*e^(d*x + c) - I)/(d*e^(7*d*x + 7*c) + 7*I*d*e^(6*d*x + 6*
c) - 21*d*e^(5*d*x + 5*c) - 35*I*d*e^(4*d*x + 4*c) + 35*d*e^(3*d*x + 3*c) + 21*I*d*e^(2*d*x + 2*c) - 7*d*e^(d*
x + c) - I*d)

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=\frac {- 140 e^{3 c} e^{3 d x} - 84 i e^{2 c} e^{2 d x} + 28 e^{c} e^{d x} + 4 i}{35 d e^{7 c} e^{7 d x} + 245 i d e^{6 c} e^{6 d x} - 735 d e^{5 c} e^{5 d x} - 1225 i d e^{4 c} e^{4 d x} + 1225 d e^{3 c} e^{3 d x} + 735 i d e^{2 c} e^{2 d x} - 245 d e^{c} e^{d x} - 35 i d} \]

[In]

integrate(1/(1-I*sinh(d*x+c))**4,x)

[Out]

(-140*exp(3*c)*exp(3*d*x) - 84*I*exp(2*c)*exp(2*d*x) + 28*exp(c)*exp(d*x) + 4*I)/(35*d*exp(7*c)*exp(7*d*x) + 2
45*I*d*exp(6*c)*exp(6*d*x) - 735*d*exp(5*c)*exp(5*d*x) - 1225*I*d*exp(4*c)*exp(4*d*x) + 1225*d*exp(3*c)*exp(3*
d*x) + 735*I*d*exp(2*c)*exp(2*d*x) - 245*d*exp(c)*exp(d*x) - 35*I*d)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (93) = 186\).

Time = 0.23 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.18 \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=\frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} + \frac {12 i \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} - \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} - \frac {4 i}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 i \, e^{\left (-2 \, d x - 2 \, c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 i \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 i \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )} - i\right )}} \]

[In]

integrate(1/(1-I*sinh(d*x+c))^4,x, algorithm="maxima")

[Out]

4/5*e^(-d*x - c)/(d*(7*e^(-d*x - c) + 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x - 3*c) - 35*I*e^(-4*d*x - 4*c) + 21
*e^(-5*d*x - 5*c) + 7*I*e^(-6*d*x - 6*c) - e^(-7*d*x - 7*c) - I)) + 12/5*I*e^(-2*d*x - 2*c)/(d*(7*e^(-d*x - c)
 + 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x - 3*c) - 35*I*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c) + 7*I*e^(-6*d*x -
 6*c) - e^(-7*d*x - 7*c) - I)) - 4*e^(-3*d*x - 3*c)/(d*(7*e^(-d*x - c) + 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x
- 3*c) - 35*I*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c) + 7*I*e^(-6*d*x - 6*c) - e^(-7*d*x - 7*c) - I)) - 4/35*I/
(d*(7*e^(-d*x - c) + 21*I*e^(-2*d*x - 2*c) - 35*e^(-3*d*x - 3*c) - 35*I*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c)
 + 7*I*e^(-6*d*x - 6*c) - e^(-7*d*x - 7*c) - I))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=-\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} + 21 i \, e^{\left (2 \, d x + 2 \, c\right )} - 7 \, e^{\left (d x + c\right )} - i\right )}}{35 \, d {\left (e^{\left (d x + c\right )} + i\right )}^{7}} \]

[In]

integrate(1/(1-I*sinh(d*x+c))^4,x, algorithm="giac")

[Out]

-4/35*(35*e^(3*d*x + 3*c) + 21*I*e^(2*d*x + 2*c) - 7*e^(d*x + c) - I)/(d*(e^(d*x + c) + I)^7)

Mupad [B] (verification not implemented)

Time = 1.68 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(1-i \sinh (c+d x))^4} \, dx=-\frac {4\,\left (21\,{\mathrm {e}}^{2\,c+2\,d\,x}-1+{\mathrm {e}}^{c+d\,x}\,7{}\mathrm {i}-{\mathrm {e}}^{3\,c+3\,d\,x}\,35{}\mathrm {i}\right )}{35\,d\,{\left (-1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^7} \]

[In]

int(1/(sinh(c + d*x)*1i - 1)^4,x)

[Out]

-(4*(exp(c + d*x)*7i + 21*exp(2*c + 2*d*x) - exp(3*c + 3*d*x)*35i - 1))/(35*d*(exp(c + d*x)*1i - 1)^7)